The Hansen solutions to the vector HHE (that can expand the free
space solutions for the vector potential or vector fields) are as
follows.
is the (normalized) elementary solution consisting of
a bessel function times
. It is (by
construction) purely transverse:
.
is the
solution constructed by the taking the curl of
.
is the
``longitudinal'' solution constructed by taking the gradient of the
scalar solution - it is left as an exercise to show that this still
satisfies the HHE. The three of these pieces span the range of possible
solutions and reconstruct an identity tensor that can be used to
construct a vector harmonic Green's function expansion.

This is summarized, with correction for factors of introduced by the
derivatives, here: